Abstract
Let n ∈ ℕ\{0, 1}. Let q be the n × n diagonal matrix with entries q11,..., qnn in] 0, +∞[. Then qℤn is a q-periodic lattice in ℝn with fundamental cell Q ≡ Πn j=0]0, qjj[. Let p ∈ Q. Let Ω be a bounded open subset of ℝn containing 0. Let G be a (nonlinear) map from ∂Ω × ℝ to ℝ. Let γ be a positive-valued function defined on a right neighbourhood of 0 in the real line. Then we consider the problem for ε > 0 small, where νp+εΩ denotes the outward unit normal to p + ε∂Ω. Under suitable assumptions and under condition limε→0+γ(ε)-1ε ∈ ℝ, we prove that the above problem has a family of solutions {u(ε, ·)}ε∈]0, ε′[ for ε′ sufficiently small, and we analyse the behaviour of such a family as ε approaches 0 by an approach which is alternative to those of asymptotic analysis.
Original language | English |
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Pages (from-to) | 511-536 |
Number of pages | 26 |
Journal | Complex Variables and Elliptic Equations |
Volume | 58 |
Issue number | 4 |
Early online date | 10 Jan 2012 |
DOIs | |
Publication status | Published - 01 Apr 2013 |
Keywords
- Laplace operator
- periodic nonlinear Robin boundary-value problem
- real-analytic continuation in Banach space
- singularly perturbed data
- singularly perturbed domain